Ina Ray Hutton, who led her all-female big band, the Melodears, all through the 1930s. Photograph: Metronome/Archive Photos

She danced like Mata Hari, she sang like Betty Boop's big sister, she shook it like a bowl of soup. And she was smart enough to work out that a top quality, all-girl jazz band would take the 1930s by storm. So how come Ina Ray Hutton, "the Blond Bombshell of Rhythm", is all but forgotten? A recently released three-disc set, The Definitive Collection, shows that she was no slouch or novelty act, either. "I'm selling this show as a music programme," she'd say with a wink, "but if curves attract an audience, so much the better."

Hutton had apple-pie looks and a jaw-dropping figure. She led her all-girl band, the Melodears, right through the 30s, popping up on screen in The Big Broadcast of 1936 alongside Bing Crosby and Al Bowlly. This was a remarkable feat when you consider there were so few other female jazz musicians in the 30s – the most visible were vibes player Margie Hyams and trumpeter Billie Rogers, both in Woody Herman's band.

Hutton was born Odessa Cowan in 1916, and grew up with her half-sister June (also a successful singer) in a black neighbourhood on Chicago's south side. US census records record her as being "negro" and "mulatto", but Hutton "passed" as white throughout her career. She studied dance, picking up a rave review in her local black newspaper, the Chicago Defender, when she was just seven, and made her Broadway debut at 14. She was 18 when the jazz impresario Irving Mills put together the all-woman band that became the Melodears and made her the leader, changing her name to Hutton to take advantage of the notorious reputation of the Woolworths heiress Barbara Hutton.

The Melodears were an instant hit, touring solidly for five years and appearing in several Paramount film shorts of their own, including the enticingly titled Feminine Rhythm (1935), Accent on Girls (1936) and Swing, Hutton, Swing (1937). The latter included the excellent Truckin', and a surviving clip reveals them to be an extremely tight and exciting band: guitarist Helen Baker keeps time by bobbing her head as Ina Ray does her best to tap dance in a tight black gown that pretty much glues her knees together. You half expect the camera to pan round and reveal a crowd of Chuck Jones's cartoon wolves, leering and cheering.

The Melodears' outfits ranged from boyish trousers to long, ultra-feminine, sequinned outfits. Downbeat magazine reported that Hutton's stage wardrobe included 400 gowns. Hutton would pave the way for a wave of female bands who took off in the 40s, when many leading male musicians were serving in the US armed forces. However, in 1939, she made the contrary decision to disband the Melodears and recruit an all-male band, including the saxophonist Serge Chaloff. Hutton was tired of her band being seen as a novelty act – reviews were uniformly snippy – and she told Downbeat she was "through with glamour". To emphasise the point, she even went brunette. It wasn't until the 1950s, by now fronting the Ina Ray Hutton Show on TV, that she got the girls back together, winning an Emmy in the process. Her last recorded performance came in the 1975 film Brother, Can You Spare a Dime? and she died in California in 1984.

One possible reason Ina Ray Hutton has been overlooked is that she cut very few records, mostly for Okeh and Elite in the early 40s; radio broadcasts make up the bulk of The Definitive Collection. Also, so many details of her story seem frustratingly buried. Interviews were rare, and there is precious little available online, but the bare bones – a platinum blond pioneer for women in music, fashion and television – cry out for a biopic. YouTube could help rebuild her reputation.

The critics may have had it in for her – luckily for us, the camera loved her.